In 1923, ten years after Bohr had derived the spectrum of atomic hydrogen by postulating the quantization of angular momentum, Louis de Broglie hit on an explanation of why the atom's angular momentum comes in multiples of ℏ.{\displaystyle \hbar .} Since 1905, Einstein had argued that electromagnetic radiation itself was quantized (and not merely its emission and absorption, as Planck held). If electromagnetic waves can behave like particles (now known as photons), de Broglie reasoned, why cannot electrons behave like waves?

Suppose that the electron in a hydrogen atom is a standing wave on what has so far been thought of as the electron's circular orbit. (The crests, troughs, and nodes of a standing wave are stationary.) For such a wave to exist on a circle, the circumference of the latter must be an integral multiple of the wavelengthλ{\displaystyle \lambda } of the former: 2πr=nλ.{\displaystyle 2\pi r=n\lambda .}

Einstein had established not only that electromagnetic radiation of frequency ν{\displaystyle \nu } comes in quanta of energy E=hν{\displaystyle E=h\nu } but also that these quanta carry a momentum p=h/λ.{\displaystyle p=h/\lambda .} Using this formula to eliminate λ{\displaystyle \lambda } from the condition 2πr=nλ,{\displaystyle 2\pi r=n\lambda ,} one obtains pr=nℏ.{\displaystyle pr=n\hbar .} But pr=mvr{\displaystyle pr=mvr} is just the angular momentum L{\displaystyle L} of a classical electron with an orbit of radius r.{\displaystyle r.} In this way de Broglie derived the condition L=nℏ{\displaystyle L=n\hbar } that Bohr had simply postulated.